How do you evaluate the expression #tan(u-v)# given #sinu=3/5# with #pi/2<u<p# and #cosv=-5/6# with #pi<v<(3pi)/2#?

How do you evaluate the expression #tan(u-v)# given #sinu=3/5# with #pi/2 < u < pi# and #cosv=-5/6# with #pi < v < (3pi)/2#?

1 Answer
Aug 5, 2018

#tan(u-v)=-(432+125sqrt11)/301#
#tan(u-v)~~-2.8126#

Explanation:

Here ,

#sinu=3/5 > 0 ,with , pi/2 < u < pito2^(nd)Quadrant#

#:.cosu=-sqrt(1-sin^2u)=-sqrt(1-9/25)=-sqrt(16/25)#

#:.cosu=-4/5#

#:.tanu=sinu/cosu=((3/5)/(-4/5))#

#:.color(blue)(tanu=-3/4#

Again ,

#cosv=-5/6 < 0,with, pi < v <(3pi)/2to3^(rd)Quadrant#

#:.sinv=-sqrt(1-cos^2v)=-sqrt(1-25/36)=-sqrt11/6#

#:.tanv=sinv/cosv=((-sqrt11/6)/(-5/6))#

#:.color(blue)(tanv=sqrt11/5#

So ,

#tan(u-v)=(tanu-tanv)/(1+tanutanv)#

#color(white)(tan(u-v))=(-3/4-sqrt11/5)/(1+(-3/4)(sqrt11/5)#

#color(white)(tan(u-v))=(-15-4sqrt11)/(20-3sqrt11#

#color(white)(tan(u-v))=(-15-4sqrt11)/(20-3sqrt11 )xx(20+3sqrt11)/(20+3sqrt11)#

#color(white)(tan(u-v))=(-300-45sqrt11-80sqrt11-132)/((20)^2-(3sqrt11)^2)#

#color(white)(tan(u-v))=(-432-125sqrt11)/(400-99)#

#tan(u-v)=-(432+125sqrt11)/301#

#tan(u-v)~~-2.8126#